Finite Group Databasebeta

← C2×#87042 #11703453

Subgroups — C2×#87042 #11703453

Order 768.

7118 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic, complemented
2#87042C2132yesmaximal, direct product factor
3C2×#5733C216yesmaximal, complemented
4C2×#5732C216yesmaximal, complemented
5C2×#5734C213yesmaximal, complemented
6C2×#5728C213yesmaximal, complemented
7C2×Q8×D12C211yesmaximal, characteristic, complemented
8C2×#5576C212yesmaximal, complemented
9C2×#5580C213yesmaximal, complemented
10C2×#5572C212yesmaximal, complemented
11C2×#5584C213yesmaximal, complemented
12C2×C4×#239C211yesmaximal, characteristic, complemented
13C2×#3518—31nomaximal, Sylow
14C22×#239C2211yescharacteristic
15C22×D3×Q8C2212yes—
16C3×Q82C22116yescomplemented
17C2×C3×C4⋊Q8C2213yes—
18C2×C3×C4⋊Q8C2216yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2213yes—
21C2×C3×C4×Q8C2212yes—
22C2×#947C2211yescharacteristic
23C2×#947C22112yes—
24C2×#946C2213yes—
25C2×#946C2216yes—
26C2×Q8×Dic3C2211yescharacteristic
27C2×Q8×Dic3C2216yes—
28#5734C22148yescomplemented
29#5733C22196yescomplemented
30#5732C22196yescomplemented
31#5728C22148yescomplemented
32#5584C22148yescomplemented
33#5580C22148yescomplemented
34#5576C22132yescomplemented
35Q8×D12C22116yescomplemented
36C4×#239C22116yescomplemented
37#5572C22132yescomplemented
38C2×#898C22112yes—
39C2×#898C2216yes—
40C2×#897C2216yes—
41C2×#897C2216yes—
42C2×#895C2213yes—
43C2×#895C2216yes—
44C2×#894C2216yes—
45C2×#893C2213yes—
46C2×#892C2213yes—
47C2×#892C2216yes—
48C2×#892C2216yes—
49C2×#890C2213yes—
50C2×#887C2216yes—
51C2×#875C2216yes—
52C2×C4×D12C2213yes—
53C2×C4×D12C2216yes—
54C2×C4×D12C2213yes—
55C2×#872C2216yes—
56C2×C42×D3C2213yes—
57C2×#869C2213yes—
58C2×C4×Dic6C2216yes—
59#3518—332no—
60C2×Q82D311yescharacteristic, complemented
61C2×#556—36no—
62C2×#556—33no—
63C2×#555—33no—
64C2×D4×Q8—31no—
65C2×#548—32no—
66C2×#542—36no—
67C2×#533—33no—
68C2×#529—33no—
69C2×#519—32no—
70C2×C4×#41—31no—
71C22×C3×Q8C2311yescharacteristic
72C22×C3×Q8C2316yes—
73C22×C3×Q8C2311yescharacteristic
74C2×#239—28no—
75C2×#239C2318yes—
76C2×D3×Q8—216no—
77C2×D3×Q8C23116yes—
78C22×D12C2313yes—
79C22×D12C2311yescharacteristic
80C22×D12C2316yes—
81C22×C4×D3C2316yes—
82C22×C4×D3C2313yes—
83C22×C4×D3C2313yes—
84C22×C4×D3C2313yes—
85C22×Dic6C2316yes—
86C3×C4⋊Q8C23124yes—
87C3×C4⋊Q8C23148yes—
88C3×C4×Q8C23116yes—
89C3×C4×Q8C2318yes—
90C3×C4×Q8C23124yes—
91C2×C3×C4⋊C4C2316yes—
92C2×C3×C4⋊C4C2313yes—
93C2×C3×C4⋊C4C2313yes—
94C2×C3×C4⋊C4C2316yes—
95C2×C3×C42C2316yes—
96C2×C3×C42C2313yes—
97#947C23196yes—
98#947C2318yes—
99#946C23148yes—
100#946C23124yes—
101Q8×Dic3C23148yes—
102Q8×Dic3C2318yes—
103C2×(C2×D3)⋊C4C2312yes—
104C2×(C2×D3)⋊C4C2316yes—
105C2×(C2×D3)⋊C4C2312yes—
106C2×(C2×D3)⋊C4C2316yes—
107C2×(C2×D3)⋊C4C2316yes—
108C2×(C2×D3)⋊C4C2316yes—
109C2×(C2×D3)⋊C4C23112yes—
110C2×C4⋊Dic3C2313yes—
111C2×C4⋊Dic3C2313yes—
112C2×C4⋊Dic3C2316yes—
113C2×C4⋊Dic3C2316yes—
114C2×Dic3⋊C4C2316yes—
115C2×Dic3⋊C4C2316yes—
116C2×C4×Dic3C23112yes—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2313yes—
119C2×C4×Dic3C2313yes—
120#898C23148yes—
121#898C23196yes—
122#897C23148yes—
123#897C23148yes—
124#895C23148yes—
125#895C23124yes—
126#894C23148yes—
127#893C23124yes—
128#892C23148yes—
129#892C23124yes—
130#892C23148yes—
131#890C23124yes—
132#887C23148yes—
133#875C23148yes—
134C4×D12C23148yes—
135C4×D12C23124yes—
136C4×D12C23124yes—
137#872C23148yes—
138C42×D3C23124yes—
139#869C23124yes—
140C4×Dic6C23148yes—
141C22×#41—31no—
142C23×Q8—32no—
143Q82C2×D3116yescomplemented
144#556—396no—
145#556—348no—
146#555—348no—
147D4×Q8—316no—
148#548—332no—
149#542—396no—
150#533—348no—
151C2×C4⋊Q8—33no—
152C2×C4⋊Q8C2×D313yes—
153C2×C4⋊Q8C2×D316yes—
154#529—348no—
155C2×#126—36no—
156C2×#126—312no—
157C2×#125—33no—
158C2×#124—31no—
159C2×#124—312no—
160C2×#124—36no—
161C2×#123—36no—
162C2×C22⋊Q8—33no—
163C2×C22⋊Q8—36no—
164C2×C22⋊Q8—36no—
165C2×C22⋊Q8—36no—
166C2×#121—33no—
167C2×#121—36no—
168#519—332no—
169C4×#41—316no—
170C2×C4×Q8—31no—
171C2×C4×Q8—36no—
172C2×C4×Q8—36no—
173C2×C4×Q8—36no—
174C2×C4×Q8C2×D311yescharacteristic
175C2×C4×Q8C2×D312yes—
176C2×C4×Q8C2×D313yes—
177C2×C4×D4—33no—
178C2×C4×D4—33no—
179C2×C4×D4—36no—
180C2×C4×D4—33no—
181C2×#117—33no—
182C2×#117—36no—
183C2×#117—36no—
184C2×#117—36no—
185C22×C42—33no—
186C23×D3C2412yes—
187C23×D3C2413yes—
188C2×C3×Q8—24no—
189C2×C3×Q8—412no—
190C2×C3×Q8#4118yes—
191C2×C3×Q8C2414yes—
192C2×C3×Q8C24124yes—
193C2×C3×Q8C2414yes—
194C22×C3×C4C2416yes—
195C22×C3×C4C2413yes—
196C22×C3×C4C2412yes—
197C22×C3×C4C2413yes—
198C22×C3×C4C2411yescharacteristic
199C22×Dic3C2413yes—
200C22×Dic3C2411yescharacteristic
201C22×Dic3C2411yescharacteristic
202C22×Dic3C2416yes—
203#239—232no—
204D3×Q8—264no—
205C2×D12—212no—
206C2×D12—24no—
207C2×D12—412no—
208C2×D12C2414yes—
209C2×D12C24112yes—
210C2×D12C24124yes—
211C2×C4×D3—224no—
212C2×C4×D3—212no—
213C2×C4×D3—212no—
214C2×C4×D3—46no—
215C2×C4×D3C24112yes—
216C2×C4×D3C24112yes—
217C2×C4×D3C24124yes—
218C2×C4×D3C24112yes—
219C2×Dic6—224no—
220C2×Dic6C24124yes—
221C3×C4⋊C4C24124yes—
222C3×C4⋊C4C24112yes—
223C3×C4⋊C4C24112yes—
224C3×C4⋊C4C24124yes—
225C3×C42C24124yes—
226C3×C42C24112yes—
227(C2×D3)⋊C4C2418yes—
228(C2×D3)⋊C4C24124yes—
229(C2×D3)⋊C4C2418yes—
230(C2×D3)⋊C4C24124yes—
231(C2×D3)⋊C4C24124yes—
232(C2×D3)⋊C4C24124yes—
233(C2×D3)⋊C4C24148yes—
234C4⋊Dic3C24112yes—
235C4⋊Dic3C24112yes—
236C4⋊Dic3C24124yes—
237C4⋊Dic3C24124yes—
238Dic3⋊C4C24124yes—
239Dic3⋊C4C24124yes—
240C4×Dic3C24148yes—
241C4×Dic3C2414yes—
242C4×Dic3C24112yes—
243C4×Dic3C24112yes—
244C2×#41—38no—
245C2×#41—68no—
246C22×Q8—316no—
247C22×Q8—36no—
248C22×Q8—616no—
249C22×Q8C22×D311yescharacteristic
250C22×Q8C22×D316yes—
251C22×Q8C22×D311yescharacteristic
252C22×D4—31no—
253C22×D4—33no—
254C22×D4—36no—
255C23×C4—36no—
256C23×C4—33no—
257C23×C4—33no—
258C23×C4—33no—
259C4⋊Q8—324no—
260C4⋊Q8C22×D3124yes—
261C4⋊Q8C22×D3148yes—
262#126—348no—
263#126—396no—
264#125—324no—
265#124—38no—
266#124—396no—
267#124—348no—
268#123—348no—
269C22⋊Q8—324no—
270C22⋊Q8—348no—
271C22⋊Q8—348no—
272C22⋊Q8—348no—
273#121—324no—
274#121—348no—
275C4×Q8—38no—
276C4×Q8—348no—
277C4×Q8—348no—
278C4×Q8—348no—
279C4×Q8C22×D318yes—
280C4×Q8C22×D3116yes—
281C4×Q8C22×D3124yes—
282C4×D4—324no—
283C4×D4—324no—
284C4×D4—348no—
285C4×D4—324no—
286#117—324no—
287#117—348no—
288#117—348no—
289#117—348no—
290C2×C4⋊C4—33no—
291C2×C4⋊C4—33no—
292C2×C4⋊C4—36no—
293C2×C4⋊C4—36no—
294C2×C4⋊C4—36no—
295C2×C4⋊C4—36no—
296C2×C4⋊C4C22×D313yes—
297C2×C4⋊C4C22×D313yes—
298C2×C4⋊C4C22×D316yes—
299C2×C4⋊C4C22×D316yes—
300C2×C22⋊C4—36no—
301C2×C22⋊C4—32no—
302C2×C22⋊C4—36no—
303C2×C22⋊C4—312no—
304C2×C22⋊C4—32no—
305C2×C22⋊C4—36no—
306C2×C22⋊C4—36no—
307C2×C42—33no—
308C2×C42—312no—
309C2×C42—33no—
310C2×C42—31no—
311C2×C42—324no—
312C2×C42C22×D316yes—
313C2×C42C22×D313yes—
314C23×C3C2511yescharacteristic
315C22×D3—24no—
316C22×D3—26no—
317C22×D3—26no—
318C22×D3—24no—
319C22×D3—26no—
320C22×D3—24no—
321C22×D3C2514yes—
322C22×D3C2516yes—
323C3×Q8—28no—
324C3×Q8—424no—
325C3×Q8C2×#41116yes—
326C2×C3×C4—24no—
327C2×C3×C4—26no—
328C2×C3×C4—212no—
329C2×C3×C4C2×#41112yes—
330C2×C3×C42−1+414yes—
331C2×C3×C4C25112yes—
332C2×C3×C4C2516yes—
333C2×C3×C4C2514yes—
334C2×C3×C4C2516yes—
335C2×C3×C4C2512yes—
336C2×Dic3—212no—
337C2×Dic3—22no—
338C2×Dic3—26no—
339C2×Dic3—22no—
340C2×Dic3C2516yes—
341C2×Dic3C2512yes—
342C2×Dic3C2512yes—
343C2×Dic3C25112yes—
344D12—224no—
345D12—28no—
346D12—424no—
347C4×D3—248no—
348C4×D3—224no—
349C4×D3—224no—
350C4×D3—412no—
351Dic6—248no—
352C24—32no—
353C24—33no—
354#41—632no—
355C2×Q8—24no—
356C2×Q8—324no—
357C2×Q8—412no—
358C2×Q8—664no—
359C2×Q8—624no—
360C2×Q8#23918yes—
361C2×Q8C23×D314yes—
362C2×Q8C23×D3124yes—
363C2×Q8C23×D314yes—
364C2×D4—34no—
365C2×D4—312no—
366C2×D4—324no—
367C2×D4—612no—
368C2×D4—64no—
369C2×D4—1212no—
370C22×C4—33no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—36no—
374C22×C4—312no—
375C22×C4—312no—
376C22×C4—324no—
377C22×C4—312no—
378C22×C4—624no—
379C22×C4—612no—
380C22×C4—612no—
381C22×C4—126no—
382C22×C4C23×D316yes—
383C22×C4C23×D313yes—
384C22×C4C23×D312yes—
385C22×C4C23×D313yes—
386C22×C4C23×D311yescharacteristic
387C4⋊C4—312no—
388C4⋊C4—312no—
389C4⋊C4—324no—
390C4⋊C4—324no—
391C4⋊C4—324no—
392C4⋊C4—324no—
393C4⋊C4C23×D3112yes—
394C4⋊C4C23×D3112yes—
395C4⋊C4C23×D3124yes—
396C4⋊C4C23×D3124yes—
397C22⋊C4—324no—
398C22⋊C4—38no—
399C22⋊C4—324no—
400C22⋊C4—348no—
401C22⋊C4—38no—
402C22⋊C4—324no—
403C22⋊C4—324no—
404C42—34no—
405C42—312no—
406C42—312no—
407C42—348no—
408C42C23×D3112yes—
409C42C23×D3124yes—
410C22×C3C22×#4112yes—
411C22×C3C2×2−1+412yes—
412C22×C3#58512yes—
413C22×C3C2611yescharacteristic
414C2×D3—24no—
415C2×D3—26no—
416C2×D3—26no—
417C2×D3—24no—
418C2×D3—26no—
419C2×D3—24no—
420C2×D3—48no—
421C2×D3—412no—
422C3×C4—24no—
423C3×C4—26no—
424C3×C4—212no—
425C3×C4C22×#41112yes—
426C3×C4C2×2−1+414yes—
427Dic3—212no—
428Dic3—22no—
429Dic3—26no—
430Dic3—22no—
431C23—34no—
432C23—36no—
433C23—64no—
434C23—66no—
435C23—66no—
436C23—64no—
437C23—66no—
438C23—64no—
439C23C24×D311yescharacteristic
440Q8—28no—
441Q8—424no—
442Q8—648no—
443Q8C2×#239116yes—
444D4—624no—
445D4—68no—
446D4—1224no—
447C2×C4—24no—
448C2×C4—26no—
449C2×C4—212no—
450C2×C4—36no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—312no—
454C2×C4—62no—
455C2×C4—612no—
456C2×C4—66no—
457C2×C4—62no—
458C2×C4—648no—
459C2×C4—624no—
460C2×C4—624no—
461C2×C4—1212no—
462C2×C4C2×#239112yes—
463C2×C4#100714yes—
464C2×C4C24×D3112yes—
465C2×C4C24×D316yes—
466C2×C4C24×D314yes—
467C2×C4C24×D316yes—
468C2×C4C24×D312yes—
469C2×C3#351814yescomplemented
470C2×C3C23×#4111yescharacteristic
471C2×C3C22×2−1+411yescharacteristic
472C2×C3C2×#58511yescharacteristic
473D3—44no—
474D3—46no—
475C22—64no—
476C22—66no—
477C22—66no—
478C22—64no—
479C22—66no—
480C22—64no—
481C22—128no—
482C22—1212no—
483C22C22×#23912yes—
484C22C2×#100712yes—
485C22#596612yes—
486C22C25×D311yescharacteristic
487C4—24no—
488C4—26no—
489C4—212no—
490C4—62no—
491C4—612no—
492C4—66no—
493C4—62no—
494C4C22×#239112yes—
495C4C2×#100714yes—
496C3C2×#351811yescharacteristic, complemented, Sylow
497C2—124no—
498C2—126no—
499C2#8704214yesdirect product factor
500C2C23×#23911yescharacteristic
501C2C22×#100711yescharacteristic
502C2C2×#596611yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×Q82C2
socleC23×C3C25
hypercenterC2×Q82D3
hypocenterC3C2×#3518
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×Q82

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×Q82C2semidirect
2. #87042C2direct (N × H)
3. C2×#5733C2semidirect
4. C2×#5732C2semidirect
5. C2×#5734C2semidirect
6. C2×#5728C2semidirect
7. C2×Q8×D12C2semidirect
8. C2×#5576C2semidirect
9. C2×#5580C2semidirect
10. C2×#5572C2semidirect
11. C2×#5584C2semidirect
12. C2×C4×#239C2semidirect
16. C3×Q82C22semidirect
28. #5734C22semidirect
29. #5733C22semidirect
30. #5732C22semidirect
31. #5728C22semidirect
32. #5584C22semidirect
33. #5580C22semidirect
34. #5576C22semidirect
35. Q8×D12C22semidirect
36. C4×#239C22semidirect
37. #5572C22semidirect
60. C2×Q82D3semidirect
143. Q82C2×D3semidirect
469. C2×C3#3518semidirect
496. C3C2×#3518semidirect
499. C2#87042direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×Q822
#870422
C2×#57332
C2×#57322
C2×#57342
C2×#57282
C2×Q8×D122
C2×#55762
C2×#55802
C2×#55722
C2×#55842
C2×C4×#2392
C2×#35183